The New Sudoku Players' Forum

[m_b_metcalf]

I got this far ... well done ... maybe that method would work !!

I'd be interested to know whether you succeed! My highly biased grid generator attempted something similar: fill in rows 1 to 8 and then try to copy them in reverse order into rows 9 to 16. The best result, after hours of trying, was given in the earlier posts.

Regards,

Mike

P.S. Another puzzle with fewer clues:

Code: Select all

        3     5                   12    14     
     6     8       11       14        1         
    10 11             16  1              6  7   
 13       16     2              7     9       12
     4           8     6 11     9          13   
  7  8          12    10 15    13           1  2
             15    13        4     2           
    16    14  3           7        6 11     9   
     9    11  6        7           3 14    16   
              2     4       13    15           
  2  1          13    15 10    12           8  7
    13           9    11  6     8           4   
 12        9     7              2    16       13
     7  6              1 16             11 10   
           1       14       11        8     6   
       14    12                    5     3        92 clues, minimal, all clues symmetrical, very hard

[coloin]

Well done ... it had to be out there !!!
Also ... been thinking about it - perhaps there is actually only one ED pallindromic solution grid !!

Edit - didnt have to look far though ...

and funnily enough it looks like Mathimagics has constructed the solution grid before !!
M A grid [most automorphisms]

[m_b_metcalf]

Also ... been thinking about it - perhaps there is actually only one ED pallindromic solution grid !!

Methinks 999_Springs' grid and Mathimagics' are different. That makes two!

Regards,

Mike

[coloin]

Yes maybe the outside ring of 12 boxes perhaps has to be fixed canonically .
and the options come from the way you fill in box 6 [not 5]], - this fixes box 11 and consequently defines the grid solution

[m_b_metcalf]

Yes maybe the outside ring of 12 boxes perhaps has to be fixed canonically .
and the options come from the way you fill in box 5, [which fixes box 11 and defines the grid solution]

Maybe. Anyway, 999_Springs' grid has 384 (6x64) UA4s and Mathimagics' 256 (4x64).

Regards,

Mike

[999_Springs]

i tried to generate palindromic 16x16 grids from scratch, by treating each grid as an 8x16 half-sudoku where the columns wrap around to the other side, i.e. r1c1...r8c1 and r1c16...r8c16 is a column, and so on.

i coded the following (naive and highly inefficient) algorithm in python from scratch

Code: Select all

fill top row with 1-16
while top half of sudoku is not filled:
    choose empty cell at random
    assign a possible candidate value for that cell at random
    do all singles
    if contradiction:
        throw the sudoku away and start again
print solution

of course using a recursion algorithm rather than throwing away an invalid sudoku would be much better, but this works for proof of concept that 16x16 palindromic solution grids are easy to find, and i couldn't quite fix the bugs in my code for the recursion, so i left it out. this found a valid grid on about 3% of iterations and i got about 2 valid grids a minute

here are 52 of them until i stopped the process after about half an hour. i replaced 10-16 by a-g for brevity. each line only represents the top half of the sudoku grid; the bottom half is, of course, the top half in reverse

palindromic 16x16 solution grids: Show

123456789abcdefg697be24ad3fgc815cg5efd9b1687243ad8fa31gc25e49b6795632a87fec1gdb4ed2164cg5b78f9a3f4g8b91e6d3a5c72bca7d5f342g961e8
123456789abcdefg6cdg3f2b785e9a14b85fa94ed3g1267ca97eg1dcf642b835d698f2g315a4cbe72b4768e5gc3d1fa9f5c37d1a29eb6g48eg1a9cb46f7852d3
123456789abcdefg8759ge126f3dacb4ae6fb9cdg5741283cgdbf43a128e7965b5fc7d9e3ga82146794eag6b5d1238cfd1g8324f7ec65ba923a6158c4bf9g7de
123456789abcdefgfd9gb4ce63218a756ac72dg35ef81b498b5e91fa4d7g36c2b81fde6c795a2g3436d97a5g28c4bf1ecg25f841db3e679a7e4a392bf1g6c85d
123456789abcdefgf8d6baecg47352195e9g43f21d867cabcb7ag19d52fe346825aedc143bg98f76791ca53b68df2g4edg8f2e67a5419bc3436b89gfec2715da
123456789abcdefgf96ed4b15g8327cacdg8a9fe24713b657ba52gc3d6ef94182gd69317f8cba54e35c14deba927gf86ba876f5g4d1ec2394e9f8a2c635gb17d
123456789abcdefge7fc4gb23d1859a6d95baecf6g72381486ag391de54f7cb2351fe29g78c6ad4bad2615fcb43ge789ceb87d6452a91g3f7g49b83a1efd26c5
123456789abcdefg756ga3ecf14d289b8db9g2f1e653c47acafe49dbg278513628a6914d3cg7b5ef5b9fec3768d2ag14d4c1f8g5a9eb7263eg736ab245f18dc9
123456789abcdefg5dbe3c1g248f769af89a2ed4167g35cb6g7c9fbad35e82142af9bde35c146g78e68ba2gc7f3d1459c315748fe9g62abd74dg1956b2a8fce3
123456789abcdefgc68fb3g4de2751a9a59efcd1g84362b7dgb7ae92f5168c3468gb192a745dcfe3e75a4gf3c18296db2cd187b6e3f9ga45f943edc5b6ag2718
123456789abcdefgd79eg2baf314c658cab6edf47g5891325g8f391c6de2a4b7367518ebd4af2gc9f8agd423e6c9751b4dc1a79528gb3fe6e92b6fcg15378d4a
123456789abcdefga8b63ge47fd129c59d57bfc234eg81a6egfcd91a6852b47321gfe49cdb8a3657beca72d5g634189f46d5813fe279gabc83796agb15cfe24d
123456789abcdefgce7fa12g38d6b459abd69efc542g713859g834bde17fca6241c9ba37268eg5dfdg2385c619f4eba7b76529efagcd1843e8fa1gd4b53729c6
123456789abcdefgcaf521g37de496b8b89eaf4d263g1c57d7g6ec9bf158a2436c7f195a4b2d38geae1gb43c687f25d959487d2g1ea3cb6f23db68fe5cg97a14
123456789abcdefgbg7cd92ef148563a9a5613fb7dge2c48edf8ga4c352619b72c1g48ed5673ba9f5e4971b2agcf3d8637ba9fc618d2eg54d68fa53g49eb721c
123456789abcdefgfgd839cb46e7125ac97efd1a5g283b64ba654g2e1f3d9c78e8ag6cbd23f971453146258fcd7bage9dc5fga976e41b8327b92e143g58acfd6
123456789abcdefgfb5c9a3edg162847d679g2cb48fe51a38egaf4d13572b69c28ce7d9564ga3f1b5dag4362fb8179ce93b18cfa7e5d62g4674fbe1gc2938d5a
123456789abcdefgec76bd1fg5428a395dfag9347e18c6b2b8g9eac2d63f5147854e7g2a6bfd391cd6b7359ce421f8gafac168db3g7925e4392g4ef158cabd76
123456789abcdefgabf59d1eg632847c7cg6b2f4d85e1a3989deacg3f417256b518b6gd729af3c4edg673feabc4891524a2c19b563edg78f3e9f482c157gabd6
123456789abcdefgd9acg4eb653f281757fb19d2e8g46ca38e6gac3fd1725b49ab58ed6cg927143f4d1ebf273ca89g562g9f384a1b567dcec67391g54feda28b
123456789abcdefgd8c6e9ga453f71b2bgf924d3e178a6c57a5e1fbc62dg8493ad857g91f6cb32e4f69142cb73edga58e37268fdg45acb19c4gb3ae52819fd76
123456789abcdefgbdfeg43a2687951ca76c9f1dg53eb482859g2cbe1fd47a36debf1a957g268c439g2a8e4fcd1367b57c83bg264e5a1d9f4615d7c3b89f2gae
123456789abcdefg7db9ga4e3f26c81586gf39c25d1ea74bcea5fd1b48g726939bfg472c138de5a6ecd7936afg5b148243281f5g6ecab97da5168ebd79423cgf
123456789abcdefgegc59d1b382f7a648a793e4fd5g61bc2fd6bacg21e473859c4de1a965b3gf287619a253ecf78g4bdb7f248dga61ec59353g8bfc7429d61ea
123456789abcdefg9gcab32df6e75148b587ef1g23d49ac6de6fa9c45g812b37a15bdge734c96f8237g864fceba219d546de329b185fg7acf92c815a6d7g34be
123456789abcdefg59d7432ae8fg61bcega8fcb12d469573b6cfed9g3517284af3219egc547ab68d6d7e85f4bcg3a9128a4gb236de91cf579cb5da17f6823ge4
123456789abcdefge56f3dcg7182ab49ca8be9f13dg42567d79ga42b56ef8c132b4eg79ca5163d8f8d791236egfb5ac45g1c8abfd43972e6a3f6d5e428c71g9b
123456789abcdefg98fb1d2c34gea576eca649fg5d1738b25gd7ae3b68f2c194f37864e2dbag591cdeg9b3c14528f6a7b5c1d8gaf679e243a642f597e1c3gbd8
123456789abcdefg59gce4daf617b2386a8eb3f12gd457c9bdf7g92ce853146ae86fa743b5cdg91275c2fe9g61a83b4d3bd918c54eg26a7f4g1a62bd73f985ec
123456789abcdefg97aefgbcd8541236cg56d1ea3f7294b8bd8f39241ge6c75a3cb7g86f294ead15f4981a35bcd72g6eda1gb24e65f83c972e659cd7g13abf84
123456789abcdefgb8d52age467f1c39f67c349d1eg82ba5aeg9bf1c253d68748d1e6cf5b39ag2473g479dbaf25e861c6c9f12437d8ga5be25ab7e8g6c143f9d
123456789abcdefg98a5gc2d3ef6b4176cb7e91fgd42a835degfba341587296c8d769egc2b1f354a35c912fa6874edgb4afg856be9d3c172eb21d3475cga869f
123456789abcdefg96be1f4d52g7a83cfg7c3a9e4d18625bda58bcg2f3e6941787gb42c61fda39e52dc1g8ea36597fb43465f91bg87e2dcae9afd357cb42g186
123456789abcdefgeb971dcg248f56a35gfc3a2e6d17b894a6d89fb453ge17c2f3g24ce1795a6bd8b7cag26fd1e834596819d753bf42gaecd45eb98acg36f217
123456789abcdefg78fcgd91635e2a4be5g9a34b72fd8c16db6afec2g81475392gb63a1e4589fd7c4c7eb2d53fg6189afa15896cdb72g4e38d9374gfaec1b265
123456789abcdefg8c6ef923g51dba47fag51ebd7634829cd7b9agc428fe31654bdg2ce6f3a71589619fg5dac48b273e3eca87f15g2964db2587349be1d6cfga
123456789abcdefgcbg5d21a76ef3984967f3gbe14d8a5c2aed849cfg5327b1654fbe72d6ga9c138e9c2683b5df1g4a7d7a6g19c384be25f3g81a5f42ec796db
123456789abcdefgd5a6bfce34g712898e7g912af5d6cb34b9cf34dg28e176a537gba2154fc89d6efc9e43671g5d28ba6d18f9ecb3a254g7245ag8bd697e3f1c
123456789abcdefg7dbf21cg46e859a389gadfeb2351c47656cea4937fdgb81294f37ab1e5862dgcd5289g46c13f7aebac16e28fdg7b9534eb7g3cd5a294168f
123456789abcdefgfcd639gb7e812a549a7bde2435fgc1865e8ga1cfd6249b37a1g7f496ed325cb8cd6ebg1254a8f97329f8e53c1b76a4gdb4537d8agfc9126e
123456789abcdefgdacgefb16854279378f5a29cde3g641b6eb934dg217f8a5c4g2b68fe3c9175dae7df9c35a6gb18428c9a21gb45d7e63f5613da47f2e8cgb9
123456789abcdefga896fdc271geb453fd5g4e9b6238c1a7c7be31ga5f4d28965e6fa2843d971cgb8b231c6de5ag7f49dcg9b7ef14865a32417a953gcbf28d6e
123456789abcdefg9ebg2ad3f64871c5c765gf412ed38a9bd8fa9ebc5g1724368b93a21g457dfc6e2age68cdb39f175474d6b5fea8c192g3f15c439762geb8da
123456789abcdefgdg86a9bc5ef37214aecfg13d248796b5975bf42e16gdca836df2459ag178eb3cbcg81ef74536ad92e5412d83bca9g76f7a93bcg6fd2e5148
123456789abcdefg89agedb2763f514cfdb7gac45e812936e65c9f31d2g487aba1d9824ec57b6fg325g37cf6a14db8e97b6fa59g38e214cd48ce31dbfg96a275
123456789abcdefg5afb9d1ce28g347689def3bg457621cacg76ae421df38b59f49g258b3ec71a6d318a49c76f2d5gbe2d67ea3f5g1b9c84be5c1g6da948f237
123456789abcdefg85fe1db36g74c9a2c96afg24e85d3b17d7gbec9a2f1368454d1573ebfc9g2a86b379681cd5a2f4geae289fdg764b15c3f6cga24513e87db9
123456789abcdefgag96d13e428f7c5bde852fcbg6731a497bfc94gade5182368a1b42d57fcge6935cdf6a1329ebg78427g3e9bf64d8a51ce6497g8c153a2bdf
123456789abcdefgeg67a24d538fc91b95cb13ef7dg48a62d8fa9cbg21e65743abd68921eg57f43c4e19g7c3a6fb28d5fc2gda564813eb978753e4fbc9d21ga6
123456789abcdefga5d6f39ceg8417b2bg8fe12ad57694c3e79cd4bg23f15a6898cb7g451d23e6af641abd8f7e5g329c7352ae16bfc98gd4defg29c346a87b15
123456789abcdefg5gbfa9c2d87e31646d79fe3b541gca82ac8e4d1g3f26597b71cg3fda4b58269ede56bcg7f2a914389af824561e3d7gbc342b189ec6g7ad5f
123456789abcdefgeb8712gc46fda5939agc3def58172b64d56f4a9b32ge178c8d4g2956abcf31e76325g8fe1479bdcaf1aeb3c7gd568942b7c9a41d23e86fg5

hope you enjoy these mike

[m_b_metcalf]

hope you enjoy these mike

Many thanks, I did. Based on the first half-grid, I produced the two puzzle below.

Regards,

Mike

Code: Select all

  1  .  .  2  .  .  3  .  4  5  .  .  6  .  .  7
  .  4  .  .  .  .  .  .  .  .  .  .  .  .  1  .
  .  .  .  8  9  .  . 10  .  .  .  3 11  .  .  .
  .  .  9  . 12  .  . 13 11  .  .  2  . 10  .  .
  .  . 14  .  .  .  .  3  9  .  .  .  .  6  .  .
  8  .  .  .  .  .  .  7 15  .  .  .  .  .  . 12
  .  2  . 16  .  .  1  8 14  6  .  . 15 13  3  .
 10  .  5  .  . 15  9  .  . 11  7  .  .  1  . 16
 16  .  1  .  .  7 11  .  .  9 15  .  .  5  . 10
  .  3 13 15  .  .  6 14  8  1  . 10 16  .  2  .
 12  .  .  .  .  .  . 15  7  .  .  .  .  .  .  8
  .  .  6  .  .  .  .  .  3  .  .  .  . 14  .  .
  .  . 10  .  2  .  . 11 13  .  . 12  .  9  .  .
  .  .  . 11  3  .  .  . 10  .  .  9  8  .  .  .
  .  1  .  .  .  .  .  .  .  .  .  .  .  .  4  .
  7  .  .  6  .  .  5  4  .  3  .  .  2  .  .  1  Extremely hard, not quite symmetric, minimal, SE = 11.6/1.2/1.2.


Code: Select all

  .  .  .  .  1  .  .  .  .  .  .  2  .  .  .  .
  .  .  .  .  .  .  .  3  4  .  .  .  .  .  .  .
  2  5  .  6  .  .  7  8  9 10  .  . 11  . 12  3
  4 13 14  . 12  .  .  2 11  .  . 15  .  8 10 16
  7  .  .  . 11  3  . 16 14  .  2  9  .  .  . 15
  .  . 11  9  . 15  .  .  .  . 16  . 14  7  .  .
  .  .  .  .  .  .  .  6 10  .  .  .  .  .  .  .
  8  2  3  .  4  1  .  .  .  .  5  7  .  9  6 13
 13  6  9  .  7  5  .  .  .  .  1  4  .  3  2  8
  .  .  .  .  .  .  . 10  6  .  .  .  .  .  .  .
  .  .  7 14  . 16  .  .  .  . 15  .  9 11  .  .
 15  .  .  .  9  2  . 14 16  .  3 11  .  .  .  7
 16 10  8  . 15  .  . 11  2  .  . 12  . 14 13  4
  3 12  . 11  .  . 10  9  8  7  .  .  6  .  5  2
  .  .  .  .  .  .  .  4  3  .  .  .  .  .  .  .
  .  .  .  .  2  .  .  .  .  .  .  1  .  .  .  .  Very easy, symmetric, not minimal, SE 2.3/1.2/1.2.

Grids renumbered to obscure order of solution values in rows 1 and 16.

[Edit: SE rating updated oncetwice.]

[coloin]

Nice work at finding those 52 grids, its difficult to say whether there will be some or all ED grid solutions ... maybe Mathimagics can tell us.

Do some have the same Box completion ?

Code: Select all

 1  2  3  4
 5  .  .  8
 9  .  . 12
13 14 15 16

but a different central 4 box completion .... [ box 6 implies 11 > completes grid].

EDIT - it might be that B1B2B3B4B5 - plus B6 - will define the palindromic grid

and another correction

only a suitable B1B2B3B5B9 with a suitable B6 defines the palindromic grid

Code: Select all

 1  2  3  .
 5  6  .  8
 9  . 11 12
 . 14 15 16

so i can forsee many 16*16 palindromic grids

[Mathimagics]

I checked, 52 grids, all ED ...

[m_b_metcalf]

Meanwhile, first update to the SE16 rating of the 1st. puzzle made above. SE16 is still running!

[coloin]

Meanwhile, first update to the SE16 rating of the 1st. puzzle made above. SE16 is still running!

Whist its running here is an 8x8 by hand

Code: Select all

+----+----+
|1234|5678|
|5678|1234|
+----+----+
|2456|....|
|3187|....|
+----+----+
|....|....|
|....|....|
+----+----+
|....|....|
|....|....|
+----+----+


Code: Select all

+----+----+
|1234|5678|
|5678|1234|
+----+----+
|2456|3187|
|3187|2456|
+----+----+
|6542|7813|
|7813|6542|
+----+----+
|4321|8765|
|8765|4321|
+----+----+


[1to9only]

Meanwhile, first update to the SE16 rating of the 1st. puzzle made above. SE16 is still running!

It is rated ED=11.6/1.2/1.2.

Solution: Show

Code: Select all

  1 11 12  2 15 14  3 16  4  5 10 13  6  8  9  7
 14  4  3 10  8 11  2  5  6 12  9  7 13 16  1 15
 13  7 15  8  9  6  4 10  1 14 16  3 11  2 12  5
  6 16  9  5 12  1  7 13 11 15  8  2  4 10 14  3
  4 15 14 12 11  5 16  3  9  8 13  1  7  6 10  2
  8  6 11  1 14  2 13  7 15 10  3 16  9  4  5 12
  9  2  7 16 10  4  1  8 14  6 12  5 15 13  3 11
 10 13  5  3  6 15  9 12  2 11  7  4 14  1  8 16
 16  8  1 14  4  7 11  2 12  9 15  6  3  5 13 10
 11  3 13 15  5 12  6 14  8  1  4 10 16  7  2  9
 12  5  4  9 16  3 10 15  7 13  2 14  1 11  6  8
  2 10  6  7  1 13  8  9  3 16  5 11 12 14 15  4
  3 14 10  4  2  8 15 11 13  7  1 12  5  9 16  6
  5 12  2 11  3 16 14  1 10  4  6  9  8 15  7 13
 15  1 16 13  7  9 12  6  5  2 11  8 10  3  4 14
  7  9  8  6 13 10  5  4 16  3 14 15  2 12 11  1


Path: Show

Code: Select all

1.2, Hidden Single: R6C11: 3 in block: r6c11=3
1.2, Hidden Single: R8C4: 3 in block: r8c4=3
1.2, Hidden Single: R11C6: 3 in block: r11c6=3
1.2, Hidden Single: R9C13: 3 in block: r9c13=3
1.5, Hidden Single: R3C9: 1 in column: r3c9=1
1.2, Hidden Single: R4C6: 1 in block: r4c6=1
1.2, Hidden Single: R14C8: 1 in block: r14c8=1
1.2, Hidden Single: R13C11: 1 in block: r13c11=1
1.5, Hidden Single: R15C9: 5 in column: r15c9=5
1.5, Hidden Single: R7C3: 7 in row: r7c3=7
1.5, Hidden Single: R10C14: 7 in row: r10c14=7
2.3, Naked Single: R1C8: 16: r1c8=16
2.3, Naked Single: R16C9: 16: r16c9=16
2.6, Pointing: Cells R6C2,R8C2: 6 in block and column: r3c2<>6, r4c2<>6
2.6, Pointing: Cells R5C15,R8C15: 8 in block and column: r1c15<>8, r4c15<>8
2.6, Pointing: Cells R9C2,R12C2: 8 in block and column: r13c2<>8, r16c2<>8
2.6, Pointing: Cells R9C15,R11C15: 6 in block and column: r13c15<>6, r14c15<>6
3.2, X-Wing: Cells R2C5,R2C12,R15C5,R15C12: 7 in 2 columns and 2 rows: r2c4<>7, r2c7<>7, r2c10<>7, r15c7<>7, r15c10<>7, r15c13<>7
3.2, X-Wing: Cells R2C3,R2C14,R15C3,R15C14: 3 in 2 columns and 2 rows: r2c1<>3, r2c16<>3, r15c1<>3, r15c16<>3
3.2, X-Wing: Cells R8C8,R9C8,R8C9,R9C9: 2 in 2 rows and 2 columns: r2c8<>2, r12c8<>2
3.2, X-Wing: Cells R7C6,R10C6,R7C11,R10C11: 12 in 2 rows and 2 columns: r1c11<>12, r2c11<>12, r3c11<>12, r5c6<>12, r5c11<>12, r12c6<>12, r12c11<>12, r14c6<>12, r15c6<>12, r16c6<>12
5.2, Jellyfish: Cells R3C1,R4C1,R3C6,R13C6,R14C6,R3C11,R4C11,R14C11,R13C16,R14C16: 6 in 4 rows and 4 columns: r2c1<>6, r2c6<>6, r2c11<>6, r6c6<>6, r11c11<>6, r15c6<>6, r15c11<>6, r15c16<>6
6.6, Turbot Fish (w/4 nodes): R12C2.9 off: r12c2<>9
7.1, Forcing Chain (w/6 nodes): R2C3.12 off: r2c3<>12
7.7, Nishio Forcing Chain (w/8 nodes): R15C1.9 on ==> R10C16.9 both on & off: r15c1<>9
8.9, Contradiction Forcing Chain (w/16 nodes): R12C7.12 on ==> R2C9.12 both on & off: r12c7<>12
9.0, Region Forcing Chains (w/17 nodes): 3 in block ==> R15C14.12 off: r15c14<>12
9.0, Region Forcing Chains (w/17 nodes): 12 in block ==> R5C10.12 off: r5c10<>12
7.1, Bidirectional Cycle (w/6 nodes): R7C11,R8C9,R8C8,R9C8,R10C6,R7C6: r12c8<>12
7.3, Forcing Chain (w/10 nodes): R10C6.9 off: r10c6<>9
2.6, Pointing: Cells R12C6,R12C8: 9 in block and row: r12c1<>9, r12c4<>9, r12c13<>9, r12c15<>9, r12c16<>9
3.2, X-Wing: Cells R7C1,R10C1,R7C16,R10C16: 9 in 2 rows and 2 columns: r2c16<>9
8.9, Region Forcing Chains (w/14 nodes): 4 in block ==> R12C10.4 off: r12c10<>4
9.0, Region Forcing Chains (w/17 nodes): 6 in block ==> R2C5.6 off: r2c5<>6
8.3, Region Forcing Chains (w/10 nodes): 4 in block ==> R11C7.4 off: r11c7<>4
8.3, Region Forcing Chains (w/10 nodes): 4 in block ==> R12C7.4 off: r12c7<>4
9.1, Region Forcing Chains (w/26 nodes): 6 in block ==> R13C1.3 on: r13c1<>4,5,14,15, r13c1=3
1.2, Hidden Single: R2C3: 3 in block: r2c3=3
1.2, Hidden Single: R4C16: 3 in block: r4c16=3
1.2, Hidden Single: R15C14: 3 in block: r15c14=3
9.0, Region Forcing Chains (w/21 nodes): 4 in block ==> R11C10.4 off: r11c10<>4
9.2, Contradiction Forcing Chain (w/45 nodes): R1C11.8 on ==> R4C11.14 both on & off: r1c11<>8
9.2, Contradiction Forcing Chain (w/47 nodes): R6C7.4 on ==> R12C11.2 both on & off: r6c7<>4
9.3, Contradiction Forcing Chain (w/56 nodes): R14C11.2 on ==> R16C3.12 both on & off: r14c11<>2
9.3, Region Forcing Chains (w/64 nodes): 4 in block ==> R6C10.4 off: r6c10<>4
9.5, Contradiction Forcing Chain (w/120 nodes): R6C4.4 on ==> R3C16.4 both on & off: r6c4<>4
9.5, Contradiction Forcing Chain (w/121 nodes): R5C4.4 on ==> R3C16.4 both on & off: r5c4<>4
9.6, Contradiction Forcing Chain (w/133 nodes): R13C16.15 on ==> R7C16.4 both on & off: r13c16<>15
9.6, Contradiction Forcing Chain (w/146 nodes): R3C6.4 on ==> R16C12.11 both on & off: r3c6<>4
2.6, Pointing: Cells R3C7,R4C7: 4 in block and column: r5c7<>4
9.7, Contradiction Forcing Chain (w/201 nodes): R5C10.4 on ==> R5C13.7 both on & off: r5c10<>4
2.8, Claiming: Cells R13C10,R14C10: 4 in column and block: r14c11<>4
9.3, Contradiction Forcing Chain (w/58 nodes): R1C11.14 on ==> R10C1.11 both on & off: r1c11<>14
9.6, Contradiction Forcing Chain (w/156 nodes): R4C1.15 on ==> R15C4.14 both on & off: r4c1<>15
9.7, Region Forcing Chains (w/38 nodes): 6 in block ==> R13C6.6 off: r13c6<>6
1.5, Hidden Single: R13C16: 6 in row: r13c16=6
9.9, Contradiction Forcing Chain (w/84 nodes): R2C1.5 on ==> R11C3.2 both on & off: r2c1<>5
9.9, Contradiction Forcing Chain (w/90 nodes): R6C13.4 on ==> R12C1.5 both on & off: r6c13<>4
9.9, Double Forcing Chain (w/92 nodes): R4C1.6 on & off ==> R4C1.14 off: r4c1<>14
9.9, Contradiction Forcing Chain (w/90 nodes): R4C11.6 on ==> R12C1.2 both on & off: r4c11<>6
1.5, Hidden Single: R4C1: 6 in row: r4c1=6
9.9, Contradiction Forcing Chain (w/94 nodes): R3C6.14 on ==> R16C12.11 both on & off: r3c6<>14
10.0, Contradiction Forcing Chain (w/109 nodes): R12C13.4 on ==> R6C4.13 both on & off: r12c13<>4
10.0, Contradiction Forcing Chain (w/113 nodes): R11C13.4 on ==> R6C4.13 both on & off: r11c13<>4
10.0, Contradiction Forcing Chain (w/124 nodes): R2C16.5 on ==> R15C11.11 both on & off: r2c16<>5
10.1, Contradiction Forcing Chain (w/145 nodes): R16C6.8 on ==> R2C9.12 both on & off: r16c6<>8
10.1, Contradiction Forcing Chain (w/151 nodes): R2C13.5 on ==> R12C12.11 both on & off: r2c13<>5
10.1, Contradiction Forcing Chain (w/170 nodes): R11C4.4 on ==> R12C13.1 both on & off: r11c4<>4
10.1, Contradiction Forcing Chain (w/180 nodes): R2C4.5 on ==> R7C16.5 both on & off: r2c4<>5
2.8, Claiming: Cells R2C5,R2C6,R2C8: 5 in row and block: r3c6<>5
10.1, Contradiction Forcing Chain (w/188 nodes): R2C5.5 on ==> R4C4.5 both on & off: r2c5<>5
10.6, Contradiction Forcing Chain (w/168 nodes): R2C6.5 on ==> R16C12.15 both on & off: r2c6<>5
1.2, Hidden Single: R2C8: 5 in block: r2c8=5
1.2, Hidden Single: R3C6: 6 in block: r3c6=6
1.5, Hidden Single: R14C11: 6 in column: r14c11=6
2.3, Naked Single: R12C8: 9: r12c8=9
10.6, Contradiction Forcing Chain (w/185 nodes): R15C13.14 on ==> R6C2.13 both on & off: r15c13<>14
10.7, Contradiction Forcing Chain (w/216 nodes): R11C7.2 on ==> R5C2.15 both on & off: r11c7<>2
10.8, Contradiction Forcing Chain (w/273 nodes): R6C5.10 on ==> R2C1.11 both on & off: r6c5<>10
10.9, Contradiction Forcing Chain (w/410 nodes): R2C4.14 on ==> R8C12.4 both on & off: r2c4<>14
9.6, Contradiction Forcing Chain (w/142 nodes): R2C11.13 on ==> R12C1.5 both on & off: r2c11<>13
11.0, Contradiction Forcing Chain (w/667 nodes): R7C11.5 on ==> R4C2.16 both on & off: r7c11<>5
11.3, Contradiction Forcing Chain (w/369 nodes): R2C12.13 on ==> R2C14.16 both on & off: r2c12<>13
11.3, Contradiction Forcing Chain (w/382 nodes): R2C11.8 on ==> R3C1.5 both on & off: r2c11<>8
11.4, Contradiction Forcing Chain (w/444 nodes): R2C13.14 on ==> R9C4.14 both on & off: r2c13<>14
10.9, Contradiction Forcing Chain (w/444 nodes): R15C5.13 on ==> R16C12.14 both on & off: r15c5<>13
11.4, Contradiction Forcing Chain (w/482 nodes): R6C7.10 on ==> R15C4.13 both on & off: r6c7<>10
11.4, Contradiction Forcing Chain (w/486 nodes): R6C10.2 on ==> R16C15.10 both on & off: r6c10<>2
11.5, Contradiction Forcing Chain (w/565 nodes): R5C10.2 on ==> R1C12.14 both on & off: r5c10<>2
11.0, Contradiction Forcing Chain (w/513 nodes): R5C5.10 on ==> R1C5.11 both on & off: r5c5<>10
11.0, Contradiction Forcing Chain (w/547 nodes): R6C6.10 on ==> R16C6.13 both on & off: r6c6<>10
11.0, Contradiction Forcing Chain (w/596 nodes): R5C6.10 on ==> R15C5.7 both on & off: r5c6<>10
11.0, Contradiction Forcing Chain (w/610 nodes): R5C7.10 on ==> R14C15.14 both on & off: r5c7<>10
2.6, Pointing: Cells R7C5,R7C6: 10 in block and row: r7c11<>10
10.8, Contradiction Forcing Chain (w/257 nodes): R7C5.5 on ==> R13C4.5 both on & off: r7c5<>5
10.8, Contradiction Forcing Chain (w/337 nodes): R12C7.2 on ==> R11C14.11 both on & off: r12c7<>2
11.0, Contradiction Forcing Chain (w/529 nodes): R10C6.5 on ==> R13C15.16 both on & off: r10c6<>5
11.5, Contradiction Forcing Chain (w/648 nodes): R11C12.4 on ==> R1C3.12 both on & off: r11c12<>4
11.5, Contradiction Forcing Chain (w/708 nodes): R1C2.15 on ==> R15C10.14 both on & off: r1c2<>15
11.6, Contradiction Forcing Chain (w/798 nodes): R15C4.14 on ==> R12C13.13 both on & off: r15c4<>14
11.6, Contradiction Forcing Chain (w/978 nodes): R9C15.12 on ==> R5C16.5 both on & off: r9c15<>12
2.6, Pointing: Cells R12C13,R12C15: 12 in block and row: r12c10<>12
2.8, Claiming: Cells R2C10,R3C10: 12 in column and block: r2c9<>12
2.0, Direct Hidden Pair: Cells R8C9,R9C9: 2,12 in column: r9c9<>6, r2c9=6
10.9, Contradiction Forcing Chain (w/504 nodes): R15C3.2 on ==> R16C15.10 both on & off: r15c3<>2
11.6, Contradiction Forcing Chain (w/901 nodes): R5C7.12 on ==> R1C12.15 both on & off: r5c7<>12
2.8, Claiming: Cells R14C7,R15C7: 12 in column and block: r15c8<>12
2.0, Direct Hidden Pair: Cells R8C8,R9C8: 2,12 in column: r8c8<>6, r15c8=6
2.8, Claiming: Cells R5C2,R5C4: 12 in row and block: r8c2<>12
11.2, Contradiction Forcing Chain (w/1392 nodes): R16C11.11 on ==> R14C7.15 both on & off: r16c11<>11
9.2, Contradiction Forcing Chain (w/42 nodes): R2C11.14 on ==> R1C15.12 both on & off: r2c11<>14
10.6, Contradiction Forcing Chain (w/147 nodes): R16C6.14 on ==> R11C13.13 both on & off: r16c6<>14
10.9, Contradiction Forcing Chain (w/404 nodes): R2C14.2 on ==> R2C13.13 both on & off: r2c14<>2
11.2, Region Forcing Chains (w/201 nodes): 13 in block ==> R2C10.13 off: r2c10<>13
11.3, Contradiction Forcing Chain (w/287 nodes): R1C15.15 on ==> R12C12.16 both on & off: r1c15<>15
11.3, Contradiction Forcing Chain (w/295 nodes): R15C11.14 on ==> R14C16.13 both on & off: r15c11<>14
11.4, Contradiction Forcing Chain (w/402 nodes): R1C15.14 on ==> R14C3.2 both on & off: r1c15<>14
11.4, Contradiction Forcing Chain (w/436 nodes): R1C2.14 on ==> R13C2.16 both on & off: r1c2<>14
11.5, Contradiction Forcing Chain (w/519 nodes): R3C7.14 on ==> R16C15.10 both on & off: r3c7<>14
11.5, Contradiction Forcing Chain (w/547 nodes): R3C2.13 on ==> R1C3.11 both on & off: r3c2<>13
11.5, Contradiction Forcing Chain (w/627 nodes): R3C10.16 on ==> R15C13.13 both on & off: r3c10<>16
11.5, Contradiction Forcing Chain (w/661 nodes): R16C5.8 on ==> R4C4.5 both on & off: r16c5<>8
11.6, Contradiction Forcing Chain (w/783 nodes): R2C10.8 on ==> R16C3.15 both on & off: r2c10<>8
11.6, Contradiction Forcing Chain (w/811 nodes): R14C10.14 on ==> R4C10.7 both on & off: r14c10<>14
11.6, Contradiction Forcing Chain (w/892 nodes): R6C13.14 on ==> R3C2.14 both on & off: r6c13<>14
11.6, Contradiction Forcing Chain (w/936 nodes): R1C12.8 on ==> R2C4.10 both on & off: r1c12<>8
10.8, Contradiction Forcing Chain (w/271 nodes): R1C14.15 on ==> R11C13.13 both on & off: r1c14<>15
10.8, Contradiction Forcing Chain (w/260 nodes): R15C7.8 on ==> R2C10.14 both on & off: r15c7<>8
11.4, Contradiction Forcing Chain (w/405 nodes): R15C6.8 on ==> R16C14.12 both on & off: r15c6<>8
11.4, Contradiction Forcing Chain (w/480 nodes): R13C10.14 on ==> R3C11.13 both on & off: r13c10<>14
11.6, Contradiction Forcing Chain (w/846 nodes): R16C15.14 on ==> R12C16.13 both on & off: r16c15<>14
11.6, Contradiction Forcing Chain (w/777 nodes): R15C12.14 on ==> R1C6.14 both on & off: r15c12<>14
11.6, Contradiction Forcing Chain (w/963 nodes): R1C11.13 on ==> R9C4.14 both on & off: r1c11<>13
9.6, Contradiction Forcing Chain (w/154 nodes): R11C12.13 on ==> R5C5.11 both on & off: r11c12<>13
9.8, Cell Forcing Chains (w/53 nodes): R11C4 ==> R11C13.9 off: r11c13<>9
9.9, Contradiction Forcing Chain (w/93 nodes): R12C5.13 on ==> R1C6.8 both on & off: r12c5<>13
11.0, Contradiction Forcing Chain (w/105 nodes): R14C2.13 on ==> R6C4.1 both on & off: r14c2<>13
11.2, Contradiction Forcing Chain (w/231 nodes): R15C6.13 on ==> R1C5.14 both on & off: r15c6<>13
11.4, Contradiction Forcing Chain (w/400 nodes): R5C2.13 on ==> R11C10.14 both on & off: r5c2<>13
10.8, Contradiction Forcing Chain (w/337 nodes): R16C2.14 on ==> R12C13.12 both on & off: r16c2<>14
11.4, Contradiction Forcing Chain (w/461 nodes): R5C15.5 on ==> R1C11.9 both on & off: r5c15<>5
11.4, Contradiction Forcing Chain (w/482 nodes): R15C7.13 on ==> R1C5.8 both on & off: r15c7<>13
11.4, Contradiction Forcing Chain (w/504 nodes): R16C15.15 on ==> R5C1.15 both on & off: r16c15<>15
10.8, Contradiction Forcing Chain (w/330 nodes): R2C13.12 on ==> R14C1.2 both on & off: r2c13<>12
9.7, Contradiction Forcing Chain (w/218 nodes): R3C15.13 on ==> R4C11.16 both on & off: r3c15<>13
10.8, Contradiction Forcing Chain (w/257 nodes): R2C16.13 on ==> R11C12.14 both on & off: r2c16<>13
10.8, Contradiction Forcing Chain (w/288 nodes): R2C5.14 on ==> R5C11.5 both on & off: r2c5<>14
11.4, Contradiction Forcing Chain (w/505 nodes): R1C15.12 on ==> R14C2.15 both on & off: r1c15<>12
3.0, Naked Pair: Cells R1C15,R2C13: 9,13 in block: r3c16<>13
10.1, Contradiction Forcing Chain (w/145 nodes): R16C2.15 on ==> R13C6.8 both on & off: r16c2<>15
10.8, Contradiction Forcing Chain (w/257 nodes): R2C10.15 on ==> R11C13.13 both on & off: r2c10<>15
10.8, Contradiction Forcing Chain (w/297 nodes): R16C2.12 on ==> R6C14.2 both on & off: r16c2<>12
9.5, Contradiction Forcing Chain (w/120 nodes): R14C14.12 on ==> R1C5.15 both on & off: r14c14<>12
10.1, Contradiction Forcing Chain (w/163 nodes): R12C5.8 on ==> R14C1.4 both on & off: r12c5<>8
10.7, Contradiction Forcing Chain (w/214 nodes): R2C14.15 on ==> R9C4.4 both on & off: r2c14<>15
10.7, Contradiction Forcing Chain (w/221 nodes): R12C5.16 on ==> R8C12.8 both on & off: r12c5<>16
10.7, Contradiction Forcing Chain (w/232 nodes): R14C10.15 on ==> R6C14.11 both on & off: r14c10<>15
10.8, Contradiction Forcing Chain (w/258 nodes): R14C1.15 on ==> R2C10.12 both on & off: r14c1<>15
10.8, Contradiction Forcing Chain (w/313 nodes): R3C3.12 on ==> R11C12.14 both on & off: r3c3<>12
9.5, Contradiction Forcing Chain (w/117 nodes): R3C15.16 on ==> R7C16.5 both on & off: r3c15<>16
10.2, Contradiction Forcing Chain (w/199 nodes): R15C1.13 on ==> R15C16.15 both on & off: r15c1<>13
10.6, Contradiction Forcing Chain (w/155 nodes): R14C7.15 on ==> R2C6.11 both on & off: r14c7<>15
10.2, Contradiction Forcing Chain (w/234 nodes): R15C16.15 on ==> R4C11.14 both on & off: r15c16<>15
10.7, Contradiction Forcing Chain (w/256 nodes): R2C14.12 on ==> R9C2.8 both on & off: r2c14<>12
9.6, Contradiction Forcing Chain (w/134 nodes): R2C10.14 on ==> R16C12.11 both on & off: r2c10<>14
10.0, Contradiction Forcing Chain (w/111 nodes): R14C2.15 on ==> R11C13.13 both on & off: r14c2<>15
10.6, Contradiction Forcing Chain (w/147 nodes): R16C14.15 on ==> R15C1.14 both on & off: r16c14<>15
10.1, Contradiction Forcing Chain (w/169 nodes): R10C16.4 on ==> R6C6.5 both on & off: r10c16<>4
10.0, Contradiction Forcing Chain (w/111 nodes): R15C16.11 on ==> R15C5.7 both on & off: r15c16<>11
2.6, Pointing: Cells R16C14,R16C15: 11 in block and row: r16c12<>11
9.5, Contradiction Forcing Chain (w/109 nodes): R13C7.16 on ==> R14C3.4 both on & off: r13c7<>16
9.6, Contradiction Forcing Chain (w/171 nodes): R15C11.8 on ==> R13C15.14 both on & off: r15c11<>8
9.8, Cell Forcing Chains (w/56 nodes): R3C14 ==> R3C16.14 off: r3c16<>14
9.8, Contradiction Forcing Chain (w/58 nodes): R13C13.14 on ==> R5C15.10 both on & off: r13c13<>14
9.5, Contradiction Forcing Chain (w/110 nodes): R7C1.4 on ==> R4C15.16 both on & off: r7c1<>4
9.6, Contradiction Forcing Chain (w/146 nodes): R14C1.13 on ==> R1C12.14 both on & off: r14c1<>13
3.4, Hidden Pair: Cells R15C4,R16C2: 9,13 in block: r15c4<>12
9.6, Contradiction Forcing Chain (w/132 nodes): R5C12.13 on ==> R12C2.7 both on & off: r5c12<>13
9.6, Contradiction Forcing Chain (w/156 nodes): R6C4.9 on ==> R12C7.8 both on & off: r6c4<>9
9.5, Contradiction Forcing Chain (w/114 nodes): R6C5.13 on ==> R16C11.14 both on & off: r6c5<>13
9.5, Contradiction Forcing Chain (w/126 nodes): R6C2.13 on ==> R16C11.14 both on & off: r6c2<>13
9.7, Contradiction Forcing Chain (w/195 nodes): R5C1.11 on ==> R11C7.10 both on & off: r5c1<>11
9.5, Contradiction Forcing Chain (w/100 nodes): R6C5.11 on ==> R16C12.14 both on & off: r6c5<>11
9.6, Contradiction Forcing Chain (w/140 nodes): R5C2.11 on ==> R15C5.8 both on & off: r5c2<>11
9.0, Contradiction Forcing Chain (w/17 nodes): R3C2.15 on ==> R1C14.12 both on & off: r3c2<>15
9.0, Region Forcing Chains (w/20 nodes): 11 in block ==> R6C14.11 off: r6c14<>11
9.2, Region Forcing Chains (w/39 nodes): 4 in row ==> R3C7.15 off: r3c7<>15
9.4, Contradiction Forcing Chain (w/79 nodes): R1C2.12 on ==> R15C10.8 both on & off: r1c2<>12
8.9, Region Forcing Chains (w/15 nodes): 16 in column ==> R14C2.16 off: r14c2<>16
9.3, Contradiction Forcing Chain (w/52 nodes): R15C10.14 on ==> R3C14.16 both on & off: r15c10<>14
2.6, Pointing: Cells R16C11,R16C12: 14 in block and row: r16c5<>14
9.4, Contradiction Forcing Chain (w/72 nodes): R3C16.15 on ==> R2C16.14 both on & off: r3c16<>15
9.4, Contradiction Forcing Chain (w/81 nodes): R3C2.16 on ==> R10C16.9 both on & off: r3c2<>16
9.4, Contradiction Forcing Chain (w/79 nodes): R3C15.15 on ==> R14C16.13 both on & off: r3c15<>15
9.5, Contradiction Forcing Chain (w/100 nodes): R3C10.15 on ==> R14C16.13 both on & off: r3c10<>15
9.5, Contradiction Forcing Chain (w/118 nodes): R6C5.5 on ==> R16C5.15 both on & off: r6c5<>5
9.5, Contradiction Forcing Chain (w/118 nodes): R6C5.16 on ==> R16C5.15 both on & off: r6c5<>16
9.5, Contradiction Forcing Chain (w/120 nodes): R3C14.4 on ==> R11C7.10 both on & off: r3c14<>4
8.3, Region Forcing Chains (w/12 nodes): 2 in row ==> R3C7.7 off: r3c7<>7
9.3, Contradiction Forcing Chain (w/49 nodes): R6C6.11 on ==> R14C16.5 both on & off: r6c6<>11
9.3, Cell Forcing Chains (w/52 nodes): R2C1 ==> R3C2.14 off: r3c2<>14
9.4, Contradiction Forcing Chain (w/75 nodes): R1C5.11 on ==> R2C4.12 both on & off: r1c5<>11
9.4, Contradiction Forcing Chain (w/82 nodes): R6C12.4 on ==> R12C1.5 both on & off: r6c12<>4
9.4, Contradiction Forcing Chain (w/88 nodes): R1C3.15 on ==> R1C11.10 both on & off: r1c3<>15
9.1, Cell Forcing Chains (w/31 nodes): R6C3 ==> R6C5.4 off: r6c5<>4
9.2, Contradiction Forcing Chain (w/39 nodes): R15C12.15 on ==> R2C14.16 both on & off: r15c12<>15
9.4, Region Forcing Chains (w/69 nodes): 11 in block ==> R2C12.8 off: r2c12<>8
2.6, Pointing: Cells R4C10,R4C11: 8 in block and row: r4c7<>8
9.2, Contradiction Forcing Chain (w/44 nodes): R11C11.14 on ==> R8C15.8 both on & off: r11c11<>14
9.3, Contradiction Forcing Chain (w/54 nodes): R3C11.14 on ==> R8C13.4 both on & off: r3c11<>14
9.4, Contradiction Forcing Chain (w/65 nodes): R4C11.14 on ==> R15C1.14 both on & off: r4c11<>14
1.5, Hidden Single: R16C11: 14 in column: r16c11=14
9.1, Region Forcing Chains (w/29 nodes): 10 in block ==> R5C12.8 off: r5c12<>8
9.2, Region Forcing Chains (w/35 nodes): 8 in column ==> R13C10.8 off: r13c10<>8
2.8, Claiming: Cells R13C6,R13C7: 8 in row and block: r15c5<>8
9.1, Contradiction Forcing Chain (w/29 nodes): R2C6.8 on ==> R13C10.15 both on & off: r2c6<>8
9.2, Contradiction Forcing Chain (w/33 nodes): R15C3.15 on ==> R2C14.8 both on & off: r15c3<>15
9.2, Region Forcing Chains (w/34 nodes): 16 in row ==> R13C6.16 off: r13c6<>16
9.1, Region Forcing Chains (w/28 nodes): 8 in row ==> R15C3.12 off: r15c3<>12
8.8, Region Forcing Chains (w/10 nodes): 12 in block ==> R15C7.15 off: r15c7<>15
9.0, Contradiction Forcing Chain (w/24 nodes): R8C12.8 on ==> R2C16.15 both on & off: r8c12<>8
1.5, Hidden Single: R8C15: 8 in row: r8c15=8
2.8, Claiming: Cells R15C12,R16C12: 8 in column and block: r15c10<>8
9.1, Cell Forcing Chains (w/29 nodes): R1C6 ==> R2C6.14 off: r2c6<>14
9.1, Region Forcing Chains (w/32 nodes): 14 in row ==> R4C7.14 off: r4c7<>14
9.1, Contradiction Forcing Chain (w/28 nodes): R2C12.15 on ==> R1C12.13 both on & off: r2c12<>15
9.1, Region Forcing Chains (w/26 nodes): 8 in column ==> R5C10.13 off: r5c10<>13
9.2, Cell Forcing Chains (w/34 nodes): R1C6 ==> R15C6.14 off: r15c6<>14
9.2, Cell Forcing Chains (w/38 nodes): R1C6 ==> R12C6.8 off: r12c6<>8
9.2, Contradiction Forcing Chain (w/40 nodes): R1C12.14 on ==> R10C16.9 both on & off: r1c12<>14
2.8, Claiming: Cells R1C5,R1C6: 14 in row and block: r2c7<>14
9.0, Cell Forcing Chains (w/19 nodes): R3C11 ==> R3C10.13 off: r3c10<>13
8.7, Cell Forcing Chains (w/42 nodes): R5C11 ==> R11C11.13 off: r11c11<>13
8.7, Cell Forcing Chains (w/42 nodes): R5C11 ==> R12C11.13 off: r12c11<>13
9.1, Cell Forcing Chains (w/32 nodes): R14C6 ==> R5C6.13 off: r5c6<>13
9.2, Contradiction Forcing Chain (w/35 nodes): R5C5.13 on ==> R5C4.1 both on & off: r5c5<>13
9.2, Contradiction Forcing Chain (w/38 nodes): R11C12.5 on ==> R6C14.4 both on & off: r11c12<>5
9.2, Contradiction Forcing Chain (w/39 nodes): R14C7.16 on ==> R14C6.14 both on & off: r14c7<>16
9.2, Contradiction Forcing Chain (w/43 nodes): R10C1.4 on ==> R1C2.13 both on & off: r10c1<>4
9.2, Contradiction Forcing Chain (w/44 nodes): R3C2.5 on ==> R13C4.4 both on & off: r3c2<>5
9.2, Region Forcing Chains (w/43 nodes): 4 in column ==> R14C1.14 off: r14c1<>14
9.2, Region Forcing Chains (w/34 nodes): 14 in column ==> R13C6.8 on: r13c6<>14, r13c6=8
9.2, Region Forcing Chains (w/35 nodes): 10 in row ==> R12C7.16 off: r12c7<>16
9.2, Contradiction Forcing Chain (w/36 nodes): R12C7.13 on ==> R12C6.10 both on & off: r12c7<>13
9.2, Region Forcing Chains (w/41 nodes): 14 in column ==> R5C13.4 off: r5c13<>4
3.4, Hidden Pair: Cells R4C13,R8C13: 4,14 in column: r4c13<>5
9.2, Contradiction Forcing Chain (w/44 nodes): R15C5.15 on ==> R6C5.14 both on & off: r15c5<>15
9.2, Region Forcing Chains (w/40 nodes): 15 in column ==> R13C15.5 off: r13c15<>5
9.2, Region Forcing Chains (w/40 nodes): 15 in column ==> R13C15.7 off: r13c15<>7
9.1, Contradiction Forcing Chain (w/32 nodes): R14C15.13 on ==> R5C11.8 both on & off: r14c15<>13
9.2, Contradiction Forcing Chain (w/34 nodes): R16C15.13 on ==> R4C11.8 both on & off: r16c15<>13
9.2, Contradiction Forcing Chain (w/35 nodes): R14C16.15 on ==> R1C15.13 both on & off: r14c16<>15
9.2, Region Forcing Chains (w/36 nodes): 8 in block ==> R5C10.16 off: r5c10<>16
9.2, Cell Forcing Chains (w/38 nodes): R14C6 ==> R1C2.13 off: r1c2<>13
8.4, Region Forcing Chains (w/16 nodes): 15 in block ==> R12C15.13 off: r12c15<>13
8.5, Region Forcing Chains (w/18 nodes): 14 in column ==> R11C12.11 off: r11c12<>11
8.5, Region Forcing Chains (w/18 nodes): 14 in column ==> R11C12.16 off: r11c12<>16
8.5, Region Forcing Chains (w/22 nodes): 15 in block ==> R12C16.11 off: r12c16<>11
8.6, Cell Forcing Chains (w/26 nodes): R11C15 ==> R12C15.11 off: r12c15<>11
8.8, Region Forcing Chains (w/11 nodes): 11 in column ==> R5C15.11 off: r5c15<>11
8.5, Cell Forcing Chains (w/18 nodes): R6C13 ==> R13C7.7 off: r13c7<>7
8.3, Cell Forcing Chains (w/12 nodes): R4C7 ==> R15C5.14 off: r15c5<>14
8.4, Cell Forcing Chains (w/14 nodes): R16C5 ==> R15C7.14 off: r15c7<>14
8.4, Region Forcing Chains (w/16 nodes): 2 in block ==> R3C7.2 off: r3c7<>2
2.3, Naked Single: R3C7: 4: r3c7=4
1.2, Hidden Single: R4C13: 4 in block: r4c13=4
1.5, Hidden Single: R8C13: 14 in column: r8c13=14
2.6, Pointing: Cells R2C6,R2C7: 2 in block and row: r2c16<>2
7.3, Forcing Chain (w/12 nodes): R5C4.13 off: r5c4<>13
7.4, Forcing Chain (w/14 nodes): R2C4.12 off: r2c4<>12
1.5, Hidden Single: R5C4: 12 in column: r5c4=12
1.2, Hidden Single: R6C4: 1 in block: r6c4=1
1.2, Hidden Single: R5C12: 1 in block: r5c12=1
1.5, Hidden Single: R2C10: 12 in row: r2c10=12
2.3, Naked Single: R5C2: 15: r5c2=15
2.6, Pointing: Cells R1C11,R2C11: 10 in block and column: r5c11<>10
3.4, Hidden Pair: Cells R1C11,R2C11: 9,10 in block: r2c11<>16
3.6, Naked Triplet: Cells R2C4,R2C11,R2C13: 9,10,13 in row: r2c1<>13
3.2, X-Wing: Cells R3C1,R3C11,R5C1,R5C11: 13 in 2 columns and 2 rows: r5c7<>13
7.2, Forcing Chain (w/8 nodes): R14C7.12 off: r14c7<>12
1.2, Hidden Single: R15C7: 12 in block: r15c7=12
1.5, Hidden Single: R12C13: 12 in column: r12c13=12
1.2, Hidden Single: R11C13: 1 in block: r11c13=1
1.2, Hidden Single: R12C5: 1 in block: r12c5=1
2.3, Naked Single: R12C15: 15: r12c15=15
1.2, Hidden Single: R14C14: 15 in block: r14c14=15
1.2, Hidden Single: R2C16: 15 in block: r2c16=15
2.6, Pointing: Cells R3C15,R4C15: 14 in block and column: r13c15<>14, r14c15<>14
2.3, Naked Single: R13C15: 16: r13c15=16
1.5, Hidden Single: R4C2: 16 in column: r4c2=16
2.3, Naked Single: R3C3: 15: r3c3=15
1.2, Hidden Single: R15C1: 15 in block: r15c1=15
1.5, Hidden Single: R15C16: 14 in row: r15c16=14
2.3, Naked Single: R4C11: 8: r4c11=8
1.2, Hidden Single: R5C10: 8 in block: r5c10=8
1.2, Hidden Single: R6C10: 10 in block: r6c10=10
2.3, Naked Single: R15C10: 2: r15c10=2
2.3, Naked Single: R15C11: 11: r15c11=11
1.2, Hidden Single: R12C12: 11 in block: r12c12=11
2.6, Pointing: Cells R10C11,R11C11,R12C11: 5 in block and column: r5c11<>5
2.8, Claiming: Cells R2C1,R3C1: 14 in column and block: r4c4<>14
2.8, Claiming: Cells R11C7,R12C7: 10 in column and block: r11c5<>10, r12c6<>10
2.8, Claiming: Cells R11C10,R12C10: 13 in column and block: r9c12<>13
2.8, Claiming: Cells R11C10,R12C10: 16 in column and block: r11c11<>16, r12c11<>16
3.4, Hidden Pair: Cells R5C13,R5C15: 7,10 in block: r5c13<>5
3.4, Hidden Pair: Cells R11C10,R12C10: 13,16 in block: r11c10<>14
2.6, Pointing: Cells R9C12,R11C12: 14 in block and column: r2c12<>14
1.5, Hidden Single: R2C1: 14 in row: r2c1=14
2.6, Pointing: Cells R1C2,R1C3: 11 in block and row: r1c6<>11
2.3, Naked Single: R1C6: 14: r1c6=14
1.5, Hidden Single: R6C5: 14 in column: r6c5=14
1.2, Hidden Single: R8C5: 6 in block: r8c5=6
1.2, Hidden Single: R6C2: 6 in block: r6c2=6
1.2, Hidden Single: R7C1: 9 in block: r7c1=9
1.2, Hidden Single: R6C3: 11 in block: r6c3=11
1.2, Hidden Single: R1C2: 11 in block: r1c2=11
1.2, Hidden Single: R2C4: 10 in block: r2c4=10
1.2, Hidden Single: R3C1: 13 in block: r3c1=13
1.2, Hidden Single: R4C4: 5 in block: r4c4=5
1.2, Hidden Single: R3C2: 7 in block: r3c2=7
1.0, Hidden Single: R1C3: 12 in block: r1c3=12
1.2, Hidden Single: R1C11: 10 in block: r1c11=10
1.2, Hidden Single: R2C11: 9 in block: r2c11=9
1.2, Hidden Single: R1C12: 13 in block: r1c12=13
1.2, Hidden Single: R4C10: 15 in block: r4c10=15
1.2, Hidden Single: R1C5: 15 in block: r1c5=15
1.2, Hidden Single: R2C12: 7 in block: r2c12=7
1.2, Hidden Single: R4C7: 7 in block: r4c7=7
1.0, Hidden Single: R4C15: 14 in row: r4c15=14
1.2, Hidden Single: R3C10: 14 in block: r3c10=14
1.0, Hidden Single: R3C11: 16 in block: r3c11=16
1.2, Hidden Single: R1C15: 9 in block: r1c15=9
1.0, Hidden Single: R1C14: 8 in row: r1c14=8
1.2, Hidden Single: R2C13: 13 in block: r2c13=13
1.2, Hidden Single: R2C14: 16 in block: r2c14=16
1.2, Hidden Single: R5C1: 4 in block: r5c1=4
1.0, Hidden Single: R8C2: 13 in block: r8c2=13
1.2, Hidden Single: R5C11: 13 in block: r5c11=13
1.2, Hidden Single: R8C9: 2 in block: r8c9=2
1.0, Hidden Single: R9C9: 12 in column: r9c9=12
1.2, Hidden Single: R7C11: 12 in block: r7c11=12
1.2, Hidden Single: R8C8: 12 in block: r8c8=12
1.0, Hidden Single: R9C8: 2 in column: r9c8=2
1.0, Hidden Single: R8C12: 4 in row: r8c12=4
1.2, Hidden Single: R6C12: 16 in block: r6c12=16
1.0, Hidden Single: R7C12: 5 in block: r7c12=5
1.2, Hidden Single: R6C13: 9 in block: r6c13=9
1.2, Hidden Single: R12C4: 7 in block: r12c4=7
1.2, Hidden Single: R10C1: 11 in block: r10c1=11
1.2, Hidden Single: R10C6: 12 in block: r10c6=12
1.2, Hidden Single: R10C16: 9 in block: r10c16=9
1.2, Hidden Single: R14C2: 12 in block: r14c2=12
1.2, Hidden Single: R15C4: 13 in block: r15c4=13
1.2, Hidden Single: R16C2: 9 in block: r16c2=9
1.2, Hidden Single: R11C4: 9 in block: r11c4=9
1.2, Hidden Single: R15C5: 7 in block: r15c5=7
1.2, Hidden Single: R15C6: 9 in block: r15c6=9
1.2, Hidden Single: R13C7: 15 in block: r13c7=15
1.2, Hidden Single: R14C7: 14 in block: r14c7=14
1.2, Hidden Single: R14C6: 16 in block: r14c6=16
1.2, Hidden Single: R15C3: 16 in block: r15c3=16
1.2, Hidden Single: R16C3: 8 in block: r16c3=8
1.2, Hidden Single: R15C12: 8 in block: r15c12=8
1.0, Hidden Single: R15C13: 10 in row: r15c13=10
1.2, Hidden Single: R5C15: 10 in block: r5c15=10
1.2, Hidden Single: R5C13: 7 in block: r5c13=7
1.0, Hidden Single: R13C13: 5 in column: r13c13=5
1.2, Hidden Single: R14C1: 5 in block: r14c1=5
1.0, Hidden Single: R12C1: 2 in column: r12c1=2
1.2, Hidden Single: R11C11: 2 in block: r11c11=2
1.2, Hidden Single: R14C3: 2 in block: r14c3=2
1.0, Hidden Single: R11C3: 4 in column: r11c3=4
1.2, Hidden Single: R12C16: 4 in block: r12c16=4
1.2, Hidden Single: R6C14: 4 in block: r6c14=4
1.2, Hidden Single: R5C16: 2 in block: r5c16=2
1.2, Hidden Single: R3C14: 2 in block: r3c14=2
1.2, Hidden Single: R3C15: 12 in block: r3c15=12
1.0, Hidden Single: R3C16: 5 in block: r3c16=5
1.2, Hidden Single: R6C15: 5 in block: r6c15=5
1.0, Hidden Single: R7C16: 11 in block: r7c16=11
1.0, Hidden Single: R14C16: 13 in column: r14c16=13
1.2, Hidden Single: R10C11: 4 in block: r10c11=4
1.0, Hidden Single: R12C11: 5 in column: r12c11=5
1.0, Hidden Single: R10C5: 5 in row: r10c5=5
1.2, Hidden Single: R5C6: 5 in block: r5c6=5
1.2, Hidden Single: R5C5: 11 in block: r5c5=11
1.0, Hidden Single: R5C7: 16 in row: r5c7=16
1.2, Hidden Single: R2C6: 11 in block: r2c6=11
1.2, Hidden Single: R2C7: 2 in block: r2c7=2
1.0, Hidden Single: R2C5: 8 in block: r2c5=8
1.2, Hidden Single: R6C6: 2 in block: r6c6=2
1.0, Hidden Single: R6C7: 13 in row: r6c7=13
1.2, Hidden Single: R11C2: 5 in block: r11c2=5
1.2, Hidden Single: R12C2: 10 in block: r12c2=10
1.2, Hidden Single: R9C2: 8 in block: r9c2=8
1.0, Hidden Single: R9C4: 14 in block: r9c4=14
1.0, Hidden Single: R13C2: 14 in column: r13c2=14
1.0, Hidden Single: R13C4: 4 in block: r13c4=4
1.0, Hidden Single: R13C10: 7 in row: r13c10=7
1.2, Hidden Single: R9C5: 4 in block: r9c5=4
1.2, Hidden Single: R7C6: 4 in block: r7c6=4
1.0, Hidden Single: R7C5: 10 in block: r7c5=10
1.2, Hidden Single: R12C7: 8 in block: r12c7=8
1.0, Hidden Single: R11C7: 10 in column: r11c7=10
1.2, Hidden Single: R11C5: 16 in block: r11c5=16
1.0, Hidden Single: R12C6: 13 in block: r12c6=13
1.0, Hidden Single: R16C5: 13 in column: r16c5=13
1.0, Hidden Single: R16C6: 10 in block: r16c6=10
1.0, Hidden Single: R12C10: 16 in row: r12c10=16
1.2, Hidden Single: R11C10: 13 in block: r11c10=13
1.0, Hidden Single: R14C10: 4 in column: r14c10=4
1.0, Hidden Single: R16C12: 15 in block: r16c12=15
1.0, Hidden Single: R14C15: 7 in row: r14c15=7
1.2, Hidden Single: R11C12: 14 in block: r11c12=14
1.0, Hidden Single: R9C12: 6 in block: r9c12=6
1.0, Hidden Single: R9C15: 13 in row: r9c15=13
1.2, Hidden Single: R11C15: 6 in block: r11c15=6
1.0, Hidden Single: R11C14: 11 in block: r11c14=11
1.0, Hidden Single: R16C14: 12 in column: r16c14=12
1.0, Hidden Single: R16C15: 11 in block: r16c15=11
ED=11.6/1.2/1.2



[Edit: 4Dec] Updated solution path to show chains length, using private version of SE16. The GitHub version does not show chains length, I'll update it at some later date.

[1to9only]

Whist its running here is an 8x8 by hand

Code: Select all

+----+----+
|1234|5678|
|5678|1234|
+----+----+
|2456|....|
|3187|....|
+----+----+
|....|....|
|....|....|
+----+----+
|....|....|
|....|....|
+----+----+


This has multiple solutions, SE listed these 2:

Hidden Text: Show

Code: Select all

12345678567812342456....3187....................................

1234567856781234245678133187245647136582682531477542836183614725
1234567856781234245673813187452687216453654387127862314543152867


[m_b_metcalf]

Meanwhile, first update to the SE16 rating of the 1st. puzzle made above. SE16 is still running!

It is rated ED=11.6/1.2/1.2 [a recent build of version 2021.9.21 on Intel i7-9700].

Many thanks. I've updated the post and dowloaded the new version, which is now rating this minimal version of the easy puzzle [Edit: done]:

Code: Select all

  .  .  .  .  1  .  .  .  .  .  .  2  .  .  .  .
  .  .  .  .  .  .  .  .  4  .  .  .  .  .  .  .
  2  5  .  6  .  .  7  8  9 10  .  . 11  . 12  .
  4 13  .  . 12  .  .  2 11  .  . 15  .  8 10 16
  7  .  .  . 11  3  . 16 14  .  2  9  .  .  . 15
  .  . 11  9  . 15  .  .  .  . 16  . 14  7  .  .
  .  .  .  .  .  .  .  6 10  .  .  .  .  .  .  .
  8  2  3  .  4  1  .  .  .  .  5  .  .  9  6 13
 13  6  9  .  .  5  .  .  .  .  1  4  .  3  2  8
  .  .  .  .  .  .  . 10  6  .  .  .  .  .  .  .
  .  .  7  .  . 16  .  .  .  .  .  .  9 11  .  .
 15  .  .  .  9  2  . 14 16  .  3 11  .  .  .  7
 16 10  8  . 15  .  . 11  2  .  . 12  .  . 13  4
  3  .  . 11  .  . 10  .  8  .  .  .  6  .  5  2
  .  .  .  .  .  .  .  4  .  .  .  .  .  .  .  .
  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .  .    Hard, non-symmetric, minimal, ED=10.8/1.2/1.2

[coloin]

Whist its running here is an 8x8 by hand

Code: Select all

+----+----+
|1234|5678|
|5678|1234|
+----+----+
|2456|....|
|3187|....|
+----+----+
|....|....|
|....|....|
+----+----+
|....|....|
|....|....|
+----+----+


This has multiple solutions......

Yes but I was demonstrating the construction of these palindromic solution grids - as in an 8x8 you only need define these 24 cells and the rest are implicit ...

just like this 6x6 can be defined by 15 cells and is presumably equivalent to one of the 4 found by 999_Springs

Code: Select all

+---+---+     +---+---+
|123|456|     |123|456|
|465|132|     |465|132|
+---+---+     +---+---+
|312|...|  -> |312|645|
|...|...|     |546|213|
+---+---+     +---+---+
|...|...|     |231|564|
|...|...|     |654|321|
+---+---+     +---+---+